To determine the horizontal compression factor of the transformation \( y = (2x)^2 \) from the parent function \( y = x^2 \), let's analyze the transformations involved.
- The original function \( y = x^2 \) has points that are symmetrical about the y-axis, such as \( (1, 1) \) and \( (-1, 1) \).
- The function \( y = (2x)^2 \) can be rewritten as \( y = 4x^2 \), which shows that it is affected both by a vertical scaling factor and a horizontal compression.
To find the horizontal compression factor, note that the "2" in the argument \( 2x \) indicates that the graph is compressed horizontally by a factor of 1/2, because for any x-coordinate, the output will be the same as that of the original quadratic at half that x-coordinate.
To help find this compression factor numerically, we can pick a point on the graph of \( y = (2x)^2 \) and compare it to the corresponding point on \( y = x^2 \):
- The point \( (1, 4) \) on \( y = (2x)^2 \) corresponds to \( x = 0.5 \) on the original function \( y = x^2 \) since: \[ y = x^2 \implies (0.5)^2 = 0.25. \]
The point \( (1, 4) \) indeed shows that when \( x = 0.5 \) in the original function, the output is 0.25, which is compressed horizontally to \( x = 1 \) producing an output of 4 for the transformed function.
Given your options, the point that most directly helps to calculate the horizontal compression factor is: Only (1, 4).
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